An approach toward the abc conjecture by means of p-adic Diophantine geometry : a new aspect via the period conjecture

ABC猜想暗示了数论中的几个著名结果。例如，莫德尔猜想(由G.Faltings解决)、Siegel-Zero问题(由A.Granville建立联系)等等。P进对数中线性形式的下界是获得与ABC猜想相似形状的非平凡上界的工具之一。这一结果是由R.Tijdeman，C.Stewart，Kunrui Yu等人处理的，虽然不足以达到猜想估计(可能的估计仍然是指数估计)，但众所周知，p-进对数中的线性bin只不过是缩短丢番图近似和ABC猜想的一个重要方法。出于这样的原因，我们对对数中的线性形式感兴趣。在这里，我们考虑一个椭圆版本，并通过G.Chudnovsky对椭圆曲线的形式群进行的方法(SInnou David和N.Hirata-Kohno，“…More Line Forms in Elliptic Logarims”，出现在Journal Fur die reine和Angewandte Mahematik，以及预印本：N.Hirata-Kohno，“p-进给椭圆对数中的线性形式”)，获得椭圆对数中的线性形式的估计。在形式文章中，解决了S.Lang的一个猜想。对椭圆对数函数的有用估计也是用SInnou David计算的：“对数函数和椭圆曲线的形式群”，见：塔塔基础研究所数学研究的丢番图方程，Narosa出版社，2008,243-256。我们采用这种方法与M.Huttner合作，得到了Gauus超几何级数的值的丢番图逼近，其中我们将超几何级数的积分表示视为在无穷点上的能动对数函数。在N.Hirata-Kohno，S.)Laishram，T.N.Shorey和R.Tijdeman的《An Euler定理的推广》中，Arithmetica学报第129(1)卷，2007年，71.102，讨论了某些指数丢番图方程解不存在的充分条件。较少

The abc conjecture implies several famous results in number theory. For example, Mordell conjecture (solved by G. Faltings), Siegel-zero problem (the connection was made by A. Granville) and so on. A lower bound for linear forms in p-adic logarithms is one tool to obtain a non-trivial upper bound which is in a similar shape of the abc conjecture. The observation was dealt by R. Tijdeman, C. Stewart, Kunrui Yu, etc. Although it is not enough to achieve the conjectural estimate (the possible estimate is still an exponential one), it is well-known that the linear bin in p-adic logarithms is nothing but an important way to abridge Diophantine approximations and the abc conjecture. Being motivated by such a reason, we are interested in linear forms in logarithms. We consider here an elliptic version and obtain an estimate for linear forms in elliptic logarithms, by means of the method of G. Chudnovsky carried out on the formal group of the elliptic curves (Sinnou David and N. Hirata-Kohno," … More Linear forms in elliptic logarithms", to appear in Journal fur die reine and angewandte Mathematik, and a preprint : N. Hirata-Kohno,"Linear forms in p-adic elliptic logarithms"). In the formal article a conjecture of S. Lang is solved. Useful estimates of elliptic logarithmic functions are also calculated with Sinnou David : "Logarithmic Functions and Formal Groups of Elliptic Curves", in : Diophantine Equations, Tata Institute of Fundamental Research, Studies in Mathematics, Narosa Publishing House, 2008, 243-256. We adapt this method to get Diophantine approximations of values of Gauus'hypergeometric series as a joint work with M. Huttner, where we regard the integral representation of the hypergeometric series as an ablelian logarithmic function at one point at the infinity.In N. Hirata-Kohno, S.)Laishram, T. N. Shorey and R. Tijdeman "An extension of a theorem of Euler", Acta Arithmetica vol. 129 (1), 2007, 71.102, sufficient conditions of non-existence of the integer solutions to certain exponential Diophantine equations are discussed. Less